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Laplace's Equation--Bispherical Coordinates

In bispherical coordinates, Laplace's equation becomes

del ^2f=((coshv-cosu)^3)/(a^2sinu){sinupartial/(partialv)(1/(coshv-cosu)(partialf)/(partialv))+partial/(partialu)((sinu)/(coshv-cosu)(partialf)/(partialu))}+((coshv-cosu)^2)/(a^2sin^2u)(partial^2f)/(partialphi^2).
(1)

Attempt separation of variables by plugging in the trial solution

f(u,v,phi)=sqrt(coshv-cosu)U(u)V(v)Psi(psi),
(2)

then divide the result by csc^2u(coshv-cosu)^(5/2) U(u)V(v)Phi(phi) to obtain

-1/4sin^2u+cosusinu(U^'(u))/(U(u))+sin^2u(U^('')(u))/(U(u)) +sin^2u(V^('')(v))/(V(v))+(Phi^('')(phi))/(Phi(phi))=0.
(3)

The function Phi(phi) then separates with

(Phi^('')(phi))/(Phi(phi))=-m^2,
(4)

giving solution

Psi(psi)=sin; cos(mphi)=sum_(k=1)^infty[A_ksin(mpsi)+B_kcos(mpsi)].
(5)

Plugging Psi(psi) back in and dividing by sin^2u gives

cotu(U^'(u))/(U(u))+(U^('')(u))/(U(u))-(m^2)/(sin^2u)-1/4+(V^('')(v))/(V(v))=0.
(6)

The function V(v) then separates with

(V^('')(v))/(V(v))=-n^2,
(7)

giving solution

V(v)=sin; cos(nv)=sum_(k=1)^infty[C_ksin(nv)+D_kcos(nv)].
(8)

Plugging V(v) back in and multiplying by V(v) gives

U^('')(u)+cotuU^'(u)-[(m^2)/(sin^2u)+(n^2+1/4)]U(u)=0,
(9)

so Laplace's equation is partially separable in bispherical coordinates. However, the Helmholtz differential equation cannot be separated in this manner.

See also

Bispherical Coordinates, Laplace's Equation

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References

Arfken, G. "Bispherical Coordinates (xi,eta,phi)." §2.14 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 115-117, 1970.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 665-666, 1953.

Cite this as:

Weisstein, Eric W. "Laplace's Equation--Bispherical Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaplacesEquationBisphericalCoordinates.html

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